Q. 4.82 - Elementary Statistics

Question

Tine Series. A collection of observations of a variable y taken at regular intervals over time is called a time series. Bocoomsic data and electrical signals are examples of time series. We can think of a time series as providing data points ${\left(x_1+y\right)}_2$where $x_0$ is the ith observation time and $y_i$is the observed value of y at time $x_i$. If a time series exhibits a linear trend, we can find that trend by determining the regression equation for the data points. We can then use the regression equation for forecasting purposes.

As an illustration, consider the data on the WeissStats site that shows the U.S. population, in millions of persons, for the years 1900 2013. as provided by the I.S. Census Beret.

a. Use the technology of your choice to lesbian a scatterplot of the data.

h. Use the technology of your choice to find the regression equation.

6. Use your result from part (b) to forecast the U.S. population for the years 2014 and 2015 .

Step-by-Step Solution

Answer

a) 

b) $-5,505+2.89\text{ Year }$

c) $\text{ Population }{\left.\right|}_{2014}=315.46\text{ million }$

$\text{ Population }{\left.\right|}_{2015}=318.35\text{ million }$

Part (a) Step 1: Given Information

To determine the scatterplot for the given data.

Part (a) Step 2: Explanation

The MINITAB can be used to create a scatterplot like follows:

- Save the data in the $x$ and $y$ columns, correspondingly.

- Select Scatter plot from the graph menu.

- Select the Scatter Plot With Connect Lines option.

- In the $y$ variables text box, type Population in the Population column.

- In the $x$ variables text box, type Year in the Year column.

- Select OK.

The scatterplot can be made as follows:

Part (b) Step 1: Given Information

To determine the regression equation.

Part (b) Step 2: Explanation

The MINITAB can be used to calculate the regression equation as follows:

- Select Stat > Regression > Regression - In the Response box, type Population.

- In Predictors, fill in the Year and Month columns.

- Select OK.

The end result will be

The regression equation is

Population $=-5,505+2.89\mathrm{Year}$

Part (c) Step 1: Given Information

To determine the regression equation.

Part (c) Step 2: Explanation

The population for the year $2014$ can be forecasted as

$$\begin{gathered} {\text{ Population }}_{2014}=-5,505+2.89\text{ Year } \\ =-5,505+2.89\left(2014\right) \\ =-5,505+5820.46 \\ =315.46\mathrm{million} \end{gathered}$$

The population for the year 2015 can be forecasted as,

$$\begin{gathered} {\text{ Population }}_{2015}=-5,505+2.89\text{ Year } \\ =-5,505+2.89\left(2015\right) \\ =-5,505+5823.35 \\ =318.35\mathrm{million} \end{gathered}$$